### 6.3 The Hayward energy

We saw that can be non-zero even in the Minkowski spacetime. This may motivate considering the
following expression
(Thus the integrand is , where is given by Equation (23).) By the Gauss equation this is
zero in flat spacetime, furthermore, it is not difficult to see that its limit at the spatial infinity is still the
ADM energy. However, using the second expression of , one can see that its limit at the future null
infinity is the Newman-Unti rather than the Bondi-Sachs energy.
In the literature there is another modification of the Hawking energy, due to Hayward [183]. His
suggestion is essentially with the only difference that the integrands above contain an additional
term, namely the square of the anholonomicity (see Sections 4.1.8 and 11.2.1). However,
we saw that is a boost gauge dependent quantity, thus the physical significance of this
suggestion is questionable unless a natural boost gauge choice, e.g. in the form of a preferred
foliation, is made. (Such a boost gauge might be that given by the main extrinsic curvature
vector and discussed in Section 4.1.2.) Although the expression for the Hayward
energy in terms of the GHP spin coefficients given in [63, 65] seems to be gauge invariant, this
is due only to an implicit gauge choice. The correct, general GHP form of the extra term is
. If, however, the GHP spinor dyad is fixed as in the large sphere or in
the small sphere calculations, then , and hence the extra term is, in fact,
.

Taking into account that near the future null infinity (see for example [338]), it is
immediate from the remark on the asymptotic behaviour of above that the Hayward energy tends to
the Newman-Unti instead of the Bondi-Sachs energy at the future null infinity. The Hayward energy
has been calculated for small spheres both in non-vacuum and vacuum [63]. In non-vacuum it
gives the expected value . However, in vacuum it is , which is
negative.